Transactions on Engineering Sciences vol 21, 1998 WIT Press, ISSN

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1 Micromechanical modelling of textile composites using variational principles A. Prodromou, Ph. Vandeurzen, G. Huysmans, J. Ivens & I. Verpoest Department ofmetallurgy and Materials Engineering, Katholieke Universiteit Leuven de Croylaan 2 B-3001 Leuven, Belgium Andreas.Prodromou@mtm. kuleuven. ac. be Abstract The ever increasing use of textile reinforced composite materials for structural applications makes the development of automated and computationally efficient analysis tools a must. In this paper, a micromechanical method for predicting the thermo-mechanical behaviour of textile reinforced (twodimensional woven and braided fabrics) composites is presented. The model considers the shape and type of the yarns, yarn interactions and crimp as well as the matrix distribution. This modelling scheme consists of a multilevel automated geometric decomposition of a representative volume element (unit cell) into smaller elements (block and micro cells) containing yarn and matrix parts. This way, the problem of stress analysis for the whole unit cell is split into a number of subproblems at each level of the decomposition scheme. This top to bottom decomposition is followed by a bottom to top homogenisation scheme in which internal stresses (sub-unit cell level) are linked to external ones (unit cell level). This procedure results in the calculation of the stress-strain field and the prediction of the elastic properties of the composite. To develop this method the complementary variational principle was used. A FORTRAN software programme called TEXCOMP-CEM, has been developed to automate calculation and turn the method into a practical design tool. This program computes stiffness, micro-stress fields and first cell failure. Results obtained are comparable to those obtained experimentally and from finite element modelling.

2 362 Computer Methods in Composite Materials 1 Introduction The anisotropic nature of textile reinforced composites makes it possible for designs to be tailored to specific loading conditions which in return results in significant gains in strength and stiffness to weight ratio. Despite these advantages their widespread use is hindered by a limited understanding of their micromechanical behaviour. The development of micromechanical models to predict the macroscopic behaviour of composites, with sufficient accuracy and efficiently, is therefore imperative. Many different models exist for modelling woven and braided fabric composites which can be grouped together in two classes. The first one includes the laminate theory and fabric geometry models[1, 2, 3, 4, 5, 6, 7]. Models in this class are primarily used to predict the thermoelastic behaviour of the composite as well as to provide a rough estimate of internal stresses. Finite element models[8, 9, 10] form the second class. In addition to providing thermoelastic analysis they also provide predictions of the internal stress state and damage analysis. Unfortunately finite element models are not computationally efficient and are, in many cases, cumbersome to use. A new modelling approach is therefore required that will provide accurate predictions of three dimensional stiffness and internal stresses in addition to being computationally efficient and user friendly. This paper presents such an approach. The model, called the Complementary Energy Model (CEM), consists of a substructuring technique used to solve the stress analysis problem of woven and braided fabric composites. The complementary variational principle (CVP)[13] has been utilised to implement this. This principle states that from all admissible stress fields the one which minimises the total complementary energy is the true one. The CVP been utilised by Aboudi[ll] and Cheng[12] for unidirectional composites. This model can predict stiffness, internal stress fields and first cell failure. The presentation that follows focuses on 2D woven fabrics but the model has also been applied to 2D braids. 2 Geometric model The model assumes an ideal one layer fabric composite. All analysis is carried out on the unit cell of the textile. The geometric analysis

3 Computer Methods in Composite Materials 363 Geometric Model Homogcnisation Stress Model] Unit-Cell [SJUnitCcll (n)unitccll (a) Unit Cell + AT 8 = LEVEL 4 LEVEL 5 Fibre Matrix [SIFihrc [SjMatrix I (n)fihrc {a)malrix {o}y;ini {ajmatrix Figure 1: Schematic representation of the Complementary Energy Model (CEM). was based on a multi-level substructuring technique which consisted of decomposing the unit cell into smaller elements (see figure 1). The unit cell (level 1) is split into block cells (level 2), micro-cells or combi-cells (level 3), matrix and yarn (level 4) and matrix and fibres layers (level 5). Geometric information necessary for the model are computed at each level. This method is computationally efficient, accurate and general in concept so that it can be applied to all 2D woven and braided fabric composites. This decomposition of the unit cell is achieved automatically with only a limited amount of input data from the weaving company (the weaving pattern) and a few measured values (composite thickness,fillyarn crimp and the yarn aspect ratio). One can consider this five level decomposition scheme as an intelligent mesh generator for 2D textile composites. 3 Elastic properties The top to bottom geometric decomposition described in the previous section is followed by a bottom to top homogenisation during

4 364 Computer Methods in Composite Materials which the elastic properties of the composite are computed. In order to calculate the 3D elastic properties of a textile composite the average stress, or strain, in each matrix and fibre phase are required. The average stress or strain in a matrix or fibre phase is related to a uniform stress on the boundary of the unit cell by concentration tensors. These fourth order tensors, A and B, are defined by the following equations. W = ^}{o} (1) {*»} - (Bi]{e} (2) a and e are the overall stress and strain tensors. The proposed multistep homogenisation technique computes these stress concentration tensors from level 5 up to level 1. It should be emphasised that no isostress, or isostrain, assumptions are made. At the two bottom levels (from 5 to 4) the elastic behaviour of the yarns is modelled using the model of Chamis[14]. The complementary variational principle is then applied, in the transition from level 4 to 3, to calculate the stress concentration tensors for the matrix and yarn layers. This also leads to the prediction of the overall combi-cell elastic behaviour. The function to be minimised is the complementary energy, U, of the combi-cell given by, u= ywl%]w (3) Sy and SM are the 3D compliance matrices for the impregnated yarn and matrix layers respectively while cr% are the layer stresses. A stress tensor is specified for each layer, {0"J^ = Mi, 0%, 03%,?232, 73H,Ti22} % = 7, M (4) Before attempting to carry out the minimisation procedure it is necessary that some constraints be set. These will provide for continuity of stresses (eqn 5) and establish a link between internal stresses and an externally applied stress tensor on the combi-cell (eqns 6 and 7) = T23M >Ty$\Y = 7"31M (5) ^12 = (7 1 ]>]feiot2i = 02 5^ ^03% = ^3 (6) = 7*23 ^] k^u = TZ\ ^ ^i^l2i = 7*12 (7)

5 Computer Methods in Composite Materials 365 The optimisation problem presented can be solved efficiently using the method of Lagrangian multipliers the end result being the stress concentration tensor for six independent external stress states. The overall compliance matrix of the combi-cell, Sec, is calculated as a function of the fractional volume k, the compliance matrix S and the concentration tensor A for each layer. In a similar fashion one can apply the same procedure all the way to level 1 the end product being the overall symmetric 3D compliance matrix of the unit cell of the textile composite. The homogenisation technique has also been extended to incorporate the calculation of thermal parameters [15]. For this case the expression for the complementary energy is given by, ^= E W%W+ E &,A%W?W (9) where AT is the temperature difference and a/% is the second order thermal expansion tensor. 4 Strength model In addition to predicting thermoelastic parameters the CEM was also used to carry out progressive failure analysis of 2D textile composites. To carry out an as realistic as possible stress analysis residual stresses were superimposed on externally applied stresses (residual stresses are a result of significant differences between the thermal expansion constants of the matrix and yarns after the part is cooled down from its processing temperature). A direct link between the external forces acting on a textile composite and cell stress at each geometric level is established by calculation of the concentration tensors. W = NM) + {DJAT (10) For the matrix material the paraboloid failure locus applied on the principal stresses (eqn ll)within the matrix cell has been utilised. 7 - <?77)2 + ((777 - (7777)2 + ((7777 -,,. 2((77 4- (777 + (7777)(% - Sr) - 2%^r > 0 (11)

6 366 Computer Methods in Composite Materials Sc is the failure stress in compression while ST is the failure strength in tension. A maximum stress criterion is used for the yarns. In this case the current yarn stresses are compared to the respective ultimate strengths. Progressive failure analysis is based on the assumption that the damaged material could be replaced with an equivalent material having degraded properties. The effects of matrix and yarn failure are taken into account in an average sense. In the GEM the stiffness reduction method proposed by Blackketter[16] is used. Blackketter's method accounts for the appropriate damage mode in the yarns by degrading the appropriate moduli when damage is detected. Additionally, matrix failure is introduced by reducing the Young and shear moduli. For the development of the strength model it was assumed that the composite is initially free of damage. The nonlinearity of the matrix is not taken into account and the model does not calculate the geometric deformation of the textile at each load step. Figure 2: Prediction of internal stresses for a carbon/dyneema weave.

7 Computer Methods in Composite Materials Model validation In order to validate model results a series of tests were carried out on a number of textile reinforced composites. Figure 2 shows the predicted out-of-plane micro-cell stresses,ct^mc, for a carbon/dyneema hybrid composite having afibrevolume fraction of 40.2%. (2x2 twill weave). A uniaxial load, as indicated in the figure, was applied. One can clearly see that a compressive stress is predicted at yarn crossover points. The two peaks were the out-of-plane stress reaches a value of 25 percent of the tensile are regions of highly undulated warp yarns. Figure 3: Comparison of experimental and theoretical stress-strain curves for a 2D plain weave composite. Figure 3 compares experimental and model stress-strain curves for the bias and warp loading directions of a glass/epoxy weave. In the case of loading in the warp direction good correlation is observed between theory and experiment. The position of the knee which is characteristic of woven fabric composites is predicted correctly by the GEM. On the other hand a discrepancy is visible in the bias loading case at high strain levels. This can be explained by the fact that the model does not account for yarn reorientation during loading which results in stress increases. As a result the ultimate bias strength is not predicted correctly.

8 368 Computer Methods in Composite Materials CEM CCM-P FGM-MX-P R3M-NX-P Relative Error % Figure 4: Relative error of the CEM and FGM based models compared to experimental results. Figure 4 shows a comparison of elastic constants obtained using the CEM and models based on the FGM[2] with experimental data. The CEM obviously presents a major improvement in prediction accuracy compared to other models. 6 Conclusions The Complementary Energy Model presented here is an addition to other micromechanical schemes used for textile composites. It can serve as a useful tool in stress analysis problems. To this end a FOR- TRAN program, TEXCOMP-CEM, has been developed to automate calculations. A drawback of this method is that accurate geometric description (orientation and position of the yarns) of the textile architecture is required. It has been applied to 2D woven and braided (extensions to 3D weaves and braids are under development) fabric composites but application to knitted materials is hindered by their complex yarn architecture. A major advantage of this model is the detailed prediction of stresses in the matrix and yarn phases as a function of the position in the unit cell, which lends itself to significantly more

9 Computer Methods in Composite Materials 369 accurate damage analysis than models belonging to the first class presented in the introduction. Acknowledgements This text presents research results of the Belgian Programme on Interuniversity Poles of Attraction, funded by the Belgian state, and of the BRITE/EURAM project MULTEXCOMP funded by the European Union. References [1] Ishikawa, T. & Chou, T.W. Stiffness and strength behaviour of woven fabric composites, Journal of Composite Materials, 17, pp , [2] Ko, F.K. & Chou, T.W. (eds.), Compose Mofenak.Senea 3- Textile Structural Composites, Elsevier Science Publishers, Amsterdam, [3] Byun, J.H, Whitney, T.J., Du, G.W. & Chou, T.W. Analytical characterisation of two-step braided composites, Journal of Composite Materials, 25, pp , [4] Naik, N.K. & Shembekar, P.S. Elastic behavior of woven fabric composites: I-Lamina analysis, Journal of Composite Materials, 26, pp , [5] Du, G.W., Chou, T.W. & Popper, P. Analysis of threedimensional textile preforms for multi-directional reinforcement of composites, Journal of Materials Science, 26, pp , [6] Pastore, C.M. & Gowayed, Y.A. A self-consistent fabric geometry model: Modification and application of a fabric geometry model to predict the elastic properties of textile composites, Journal of Composites Technology & Research, 16, pp , [7] Varideurzen, Ph., Ivens, J. & Verpoest, I. Need title here, Composites Science and Technology, 56, pp , 1996.

10 370 Computer Methods in Composite Materials [8] Paumelle, P., Hassim, A. & Lene, F. Composites with woven reinforcements: calculation and Parametric analysis of the properties of homogeneous equivalent, La Recherche Aerospatiale, 1, pp. 1-12,1990. [9] Paumelle, P., Hassim, A. & Lene, F. Microstress analysis in woven composite structures, La Recherche Aerospatiale, 6, pp , [10] Whitcomb, J., Woo, K. & Gundapaneni, S. Macro finite element analysis of textile composites, Journal of Composite Materials, 7, pp , [11] Aboudi, J. Micromechanical analysis of composites by the method of cells, Applied Mechanics Review, 42, pp , [12] Chen, D. & Cheng, S. Analysis of composite materials: A micromechanical approach, Journal of Reinforced Plastics and Composites, 12, pp , [13] Leech, C.M. International Journal of Mechanical Engineering Education, 9, pp , [14] Chamis, C., Simplified composite micromechanics equations for hygral, thermal and mechanical properties, Proc. of the 38th Conf. of the Society of the Plastics Industry, pp. 7-11, [15] Vandeurzen, Ph., Ivens, J. & Verpoest I. Micro-stress analysis in woven fabric composites by variational principles, Proc. Conf. Deformation and Fracture of Composite Materials, ed. J. Hodgkinson pp , [16] Blackketter, D.M., Walrath, D.E. & Hansen, A.C. Modeling damage in plain weave fabric reinforced composite materials, Journal of Composites Technology and Research, 15, pp , 1993.